| L(s) = 1 | + 3-s + 2·5-s + 5·7-s − 2·11-s + 2·13-s + 2·15-s + 19-s + 5·21-s + 5·25-s − 27-s + 8·29-s + 9·31-s − 2·33-s + 10·35-s − 3·37-s + 2·39-s − 20·41-s − 10·43-s − 6·47-s + 18·49-s − 12·53-s − 4·55-s + 57-s − 12·59-s − 10·61-s + 4·65-s − 5·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.88·7-s − 0.603·11-s + 0.554·13-s + 0.516·15-s + 0.229·19-s + 1.09·21-s + 25-s − 0.192·27-s + 1.48·29-s + 1.61·31-s − 0.348·33-s + 1.69·35-s − 0.493·37-s + 0.320·39-s − 3.12·41-s − 1.52·43-s − 0.875·47-s + 18/7·49-s − 1.64·53-s − 0.539·55-s + 0.132·57-s − 1.56·59-s − 1.28·61-s + 0.496·65-s − 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.764954788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.764954788\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.61939431803824956055912989940, −11.40600450184990641011993417741, −10.64833204944929744237983947565, −10.58122589564776834266931025629, −9.807703268306811534641644660209, −9.683132609076294109829606803332, −8.596179846730826081126212669275, −8.575184208198096266151663157800, −8.163797806184969046578832758502, −7.80582835615406115758091385680, −6.80391003450114410908091692358, −6.69096001391946685916152534534, −5.82661766187358698849374095707, −5.26448461159680183834924613663, −4.69813009178533278507597604460, −4.56022737392867558812610682089, −3.22664450895118351570326821253, −2.91328961523927284058033257501, −1.79967279993630797846841592914, −1.47431717086110302589789240265,
1.47431717086110302589789240265, 1.79967279993630797846841592914, 2.91328961523927284058033257501, 3.22664450895118351570326821253, 4.56022737392867558812610682089, 4.69813009178533278507597604460, 5.26448461159680183834924613663, 5.82661766187358698849374095707, 6.69096001391946685916152534534, 6.80391003450114410908091692358, 7.80582835615406115758091385680, 8.163797806184969046578832758502, 8.575184208198096266151663157800, 8.596179846730826081126212669275, 9.683132609076294109829606803332, 9.807703268306811534641644660209, 10.58122589564776834266931025629, 10.64833204944929744237983947565, 11.40600450184990641011993417741, 11.61939431803824956055912989940
